Triangular cross section
Finally, we consider the torsion of a cylinder with equilateral triangle cross section, as shown in figure 9.21. The boundary of this section is defined by,
where a is a constant and we have simply used the product form of each boundary line equation. Assuming that the Prandtl stress function to be of the form,
so that the boundary condition (9.50) is satisfied. It can then be verified that the potential given in equation (9.99) satisfies (9.47) if
Substituting equation (9.99) in (9.57) and using (9.100) we obtain the torque to be
where the polar moment of inertia for the equilateral triangle section, J = 3 √ 3a4 .
The shear stresses given in equation (9.44) evaluates to
on using equations (9.99) and (9.100). The magnitude of the shear stress at any point is given by,
Since, for torsion the maximum shear stress occurs at the boundary of the cross section, we investigate the same at the three boundary lines. We begin with the boundary x = a. It is evident from (9.102) that on this boundary σxz = 0. Then, it follows from (9.103) that σyz is maximum at y = 0 and this maximum value is 3aµΩ/2. It can be seen that on the other two boundaries too the maximum shear stress, τmax = 3aµΩ/2. Substituting (9.99) in equations (9.45) and (9.46) and solving the first order differential equations, we obtain the warping displacement as,
on using the condition that the origin of the coordinate system does not get displaced; a requirement to prevent the body from displacing as a rigid body.
Figure 9.22: Example of a multiply connected cross section. Elliptical cross section with two circular holes.